Lectures on Vertex Operator Algebras and Conformal Blocks

These are the lecture notes for a course taught at Tsinghua University in the spring of 2022. In these notes, we develop the basic theory of vertex operator algebras (VOAs) and their conformal blocks using complex-analytic methods. In particular, many well-known subtleties in VOA theory (about formal variables and, e.g., delta-functions) are presented in the form of proving the absolute and locally uniform (a.l.u.) convergence of certain series of complex analytic functions. We also provide many motivations from the perspective of Segal CFT.